Quantum computational chemistry

• 1929: Dirac noted the inherent complexity of quantum mechanical equations, underscoring the difficulties in solving these equations using classical computation.{{Cite journal |date=1929-04-06 |title=Quantum mechanics of many-electron systems |journal=Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character |language=en |volume=123 |issue=792 |pages=714–733 |doi=10.1098/rspa.1929.0094 |bibcode=1929RSPSA.123..714D |issn=0950-1207 |last1=Dirac |first1=P. A. M. |s2cid=121992478 |doi-access=free }}
• 1982: Feynman proposed using quantum hardware for simulations, addressing the inefficiency of classical computers in simulating quantum systems.{{Cite book |last=Feynman |first=Richard P. |editor-first1=Tony |editor-first2=Robin W. |editor-last1=Hey |editor-last2=Allen |url=https://www.taylorfrancis.com/books/mono/10.1201/9780429500442/feynman-lectures-computation-richard-feynman |title=Feynman Lectures On Computation |date=2019-06-17 |publisher=CRC Press |isbn=978-0-429-50044-2 |location=Boca Raton |doi=10.1201/9780429500442|s2cid=53898623 }}

While there are several common methods in quantum chemistry, the section below lists only a few examples.

{{Main|Unitary transformation (quantum mechanics)}}

Qubitization is a mathematical and algorithmic concept in quantum computing for the simulation of quantum systems via Hamiltonian dynamics. The core idea of qubitization is to encode the problem of Hamiltonian simulation in a way that is more efficiently processable by quantum algorithms.{{Cite journal |last1=Low |first1=Guang Hao |last2=Chuang |first2=Isaac L. |date=2019-07-12 |title=Hamiltonian Simulation by Qubitization |url=https://quantum-journal.org/papers/q-2019-07-12-163/ |journal=Quantum |language=en-GB |volume=3 |page=163 |doi=10.22331/q-2019-07-12-163|bibcode=2019Quant...3..163L |s2cid=119109921 |doi-access=free |arxiv=1610.06546 }}

Qubitization involves a transformation of the Hamiltonian operator, a central object in quantum mechanics representing the total energy of a system. In classical computational terms, a Hamiltonian can be thought of as a matrix describing the energy interactions within a quantum system. The goal of qubitization is to embed this Hamiltonian into a larger, unitary operator, which is a type of operator in quantum mechanics that preserves the norm of vectors upon which it acts.

Mathematically, the process of qubitization constructs a unitary operator $U$ such that a specific projection of $U$ is proportional to the Hamiltonian $H$ of interest. This relationship can often be represented as $H\; =\; \backslash langle\; G\; |\; U\; |\; G\; \backslash rangle$, where  $|\; G\; \backslash rangle$ is a specific quantum state and $\backslash langle\; G\; |$ is its conjugate transpose. The efficiency of this method comes from the fact that the unitary operator $U$ can be implemented on a quantum computer with fewer resources (like qubits and quantum gates) than would be required for directly simulating $H.$

A key feature of qubitization is in simulating Hamiltonian dynamics with high precision while reducing the quantum resource overhead. This efficiency is especially beneficial in quantum algorithms where the simulation of complex quantum systems is necessary, such as in quantum chemistry and materials science simulations. Qubitization also develops quantum algorithms for solving certain types of problems more efficiently than classical algorithms. For instance, it has implications for the Quantum Phase Estimation algorithm, which is fundamental in various quantum computing applications, including factoring and solving linear systems of equations.

In Gaussian orbital basis sets, phase estimation algorithms have been optimized empirically from $\backslash mathcal\{O\}(M^\{11\})$ to $\backslash mathcal\{O\}(M^\{5\})$ where $M$ is the number of basis sets. Advanced Hamiltonian simulation algorithms have further reduced the scaling, with the introduction of techniques like Taylor series methods and qubitization, providing more efficient algorithms with reduced computational requirements.{{Cite journal |last1=Kwon |first1=Hyuk-Yong |last2=Curtin |first2=Gregory M. |last3=Morrow |first3=Zachary |last4=Kelley |first4=C. T. |last5=Jakubikova |first5=Elena |date=2023-07-15 |title=Adaptive basis sets for practical quantum computing |journal=International Journal of Quantum Chemistry |language=en |volume=123 |issue=14 |doi=10.1002/qua.27123 |s2cid=253510818 |issn=0020-7608|doi-access=free |arxiv=2211.06471 }}

Plane wave basis sets, suitable for periodic systems, have also seen advancements in algorithm efficiency, with improvements in product formula-based approaches and Taylor series methods.

{{see also|Quantum fourier transform|Quantum phase estimation algorithm}}

Phase estimation, as proposed by Kitaev in 1996, identifies the lowest energy eigenstate ( $|\; E\_0\; \backslash rangle$ ) and excited states ( $|\; E\_i\; \backslash rangle$ ) of a physical Hamiltonian, as detailed by Abrams and Lloyd in 1999.{{Cite report |url=https://eccc.weizmann.ac.il//report/1996/003/ |title=Quantum measurements and the Abelian Stabilizer Problem |last=Kitaev |first=Alexei |date=1996-01-17 |publisher=Electronic Colloquium on Computational Complexity (ECCC) |issue=TR96-003 |language=en}} In quantum computational chemistry, this technique is employed to encode fermionic Hamiltonians into a qubit framework.{{Cite journal |last1=Abrams |first1=Daniel S. |last2=Lloyd |first2=Seth |date=1999-12-13 |title=Quantum Algorithm Providing Exponential Speed Increase for Finding Eigenvalues and Eigenvectors |url=https://link.aps.org/doi/10.1103/PhysRevLett.83.5162 |journal=Physical Review Letters |volume=83 |issue=24 |pages=5162–5165 |arxiv=quant-ph/9807070 |bibcode=1999PhRvL..83.5162A |doi=10.1103/PhysRevLett.83.5162 |s2cid=118937256}}

File:Quantum phase estimation steps.png

The qubit register is initialized in a state, which has a nonzero overlap with the Full Configuration Interaction (FCI) target eigenstate of the system. This state $|\; \backslash psi\; \backslash rangle$ is expressed as a sum over the energy eigenstates of the Hamiltonian, $|\backslash psi\; \backslash rangle\; =\; \backslash sum\_\{i\; =\; 1\}\; c\_i\; |E\_i\; \backslash rangle$ , where $c\_i$represents complex coefficients.{{Cite book |last1=Nielsen |first1=Michael A. |title=Quantum computation and quantum information |last2=Chuang |first2=Isaac L. |date=2010 |publisher=Cambridge university press |isbn=978-1-107-00217-3 |edition=10th anniversary |location=Cambridge}}

Each ancilla qubit undergoes a Hadamard gate application, placing the ancilla register in a superposed state. Subsequently, controlled gates, as shown above, modify this state.

This transform is applied to the ancilla qubits, revealing the phase information that encodes the energy eigenvalues.

The ancilla qubits are measured in the Z basis, collapsing the main register into the corresponding energy eigenstate $|\; E\_i\; \backslash rangle$ based on the probability $|c\_i|^2$.

The algorithm requires $\backslash omega$ ancilla qubits, with their number determined by the desired precision and success probability of the energy estimate. Obtaining a binary energy estimate precise to n bits with a success probability $p$ necessitates.$\backslash omega\; =\; n\; +\; \backslash lceil\; \backslash log\_2\; \backslash left(2\; +\; \backslash frac\{1\}\{2p\}\backslash right)\; \backslash rceil$ ancilla qubits. This phase estimation has been validated experimentally across various quantum architectures.

The total coherent time evolution $T$ required for the algorithm is approximately $T\; =\; 2^\{(\backslash omega\; +\; 1)\}\backslash pi$.{{Cite journal |last1=Du |first1=Jiangfeng |last2=Xu |first2=Nanyang |last3=Peng |first3=Xinhua |last4=Wang |first4=Pengfei |last5=Wu |first5=Sanfeng |last6=Lu |first6=Dawei |date=2010-01-22 |title=NMR Implementation of a Molecular Hydrogen Quantum Simulation with Adiabatic State Preparation |url=https://link.aps.org/doi/10.1103/PhysRevLett.104.030502 |journal=Physical Review Letters |language=en |volume=104 |issue=3 |page=030502 |doi=10.1103/PhysRevLett.104.030502 |pmid=20366636 |bibcode=2010PhRvL.104c0502D |issn=0031-9007|url-access=subscription }} The total evolution time is related to the binary precision $\backslash varepsilon\_\{\backslash text\{PE\}\}\; =\; \backslash frac\{1\}\{2^n\}$, with an expected repeat of the procedure for accurate ground state estimation. Errors in the algorithm include errors in energy eigenvalue estimation ($\backslash varepsilon\_\{PE\}$), unitary evolutions ($\backslash varepsilon\_\{U\}$), and circuit synthesis errors ($\backslash varepsilon\_\{CS\}$), which can be quantified using techniques like the Solovay-Kitaev theorem.{{Cite journal |last1=Lanyon |first1=B. P. |last2=Whitfield |first2=J. D. |last3=Gillett |first3=G. G. |last4=Goggin |first4=M. E. |last5=Almeida |first5=M. P. |last6=Kassal |first6=I. |last7=Biamonte |first7=J. D. |last8=Mohseni |first8=M. |last9=Powell |first9=B. J. |last10=Barbieri |first10=M. |last11=Aspuru-Guzik |first11=A. |last12=White |first12=A. G. |date=2010 |title=Towards quantum chemistry on a quantum computer |url=https://www.nature.com/articles/nchem.483 |journal=Nature Chemistry |language=en |volume=2 |issue=2 |pages=106–111 |doi=10.1038/nchem.483 |pmid=21124400 |arxiv=0905.0887 |bibcode=2010NatCh...2..106L |s2cid=640752 |issn=1755-4349}}

The phase estimation algorithm can be enhanced or altered in several ways, such as using a single ancilla qubit  for sequential measurements, increasing efficiency, parallelization, or enhancing noise resilience in analytical chemistry. The algorithm can also be scaled using classically obtained knowledge about energy gaps between states.{{Cite journal |arxiv=2209.14278 |first1=Youle |last1=Wang |first2=Lei |last2=Zhang |title=Quantum Phase Processing and its Applications in Estimating Phase and Entropies |date=2023 |last3=Yu |first3=Zhan |last4=Wang |first4=Xin|journal=Physical Review A |volume=108 |issue=6 |page=062413 |doi=10.1103/PhysRevA.108.062413 |bibcode=2023PhRvA.108f2413W }}

Effective state preparation is needed, as a randomly chosen state would exponentially decrease the probability of collapsing to the desired ground state. Various methods for state preparation have been proposed, including classical approaches and quantum techniques like adiabatic state preparation.{{Cite journal |last1=Sugisaki |first1=Kenji |last2=Toyota |first2=Kazuo |last3=Sato |first3=Kazunobu |last4=Shiomi |first4=Daisuke |last5=Takui |first5=Takeji |date=2022-07-25 |title=Adiabatic state preparation of correlated wave functions with nonlinear scheduling functions and broken-symmetry wave functions |journal=Communications Chemistry |language=en |volume=5 |issue=1 |page=84 |doi=10.1038/s42004-022-00701-8 |issn=2399-3669 |pmc=9814591 |pmid=36698020}}

{{Main|Variational quantum eigensolver}}

The variational quantum eigensolver is an algorithm in quantum computing, crucial for near-term quantum hardware.{{Cite journal |last1=Peruzzo |first1=Alberto |last2=McClean |first2=Jarrod |last3=Shadbolt |first3=Peter |last4=Yung |first4=Man-Hong |last5=Zhou |first5=Xiao-Qi |last6=Love |first6=Peter J. |last7=Aspuru-Guzik |first7=Alán |last8=O'Brien |first8=Jeremy L. |date=2014-07-23 |title=A variational eigenvalue solver on a photonic quantum processor |journal=Nature Communications |language=en |volume=5 |issue=1 |page=4213 |doi=10.1038/ncomms5213 |issn=2041-1723 |pmc=4124861 |pmid=25055053|arxiv=1304.3061 |bibcode=2014NatCo...5.4213P }} Initially proposed by Peruzzo et al. in 2014 and further developed by McClean et al. in 2016, VQE finds the lowest eigenvalue of Hamiltonians, particularly those in chemical systems.{{Cite journal |last1=Peruzzo |first1=Alberto |last2=McClean |first2=Jarrod |last3=Shadbolt |first3=Peter |last4=Yung |first4=Man-Hong |last5=Zhou |first5=Xiao-Qi |last6=Love |first6=Peter J. |last7=Aspuru-Guzik |first7=Alán |last8=O’Brien |first8=Jeremy L. |date=2014-07-23 |title=A variational eigenvalue solver on a photonic quantum processor |journal=Nature Communications |language=en |volume=5 |issue=1 |pages=4213 |doi=10.1038/ncomms5213 |issn=2041-1723 |pmc=4124861 |pmid=25055053|arxiv=1304.3061 |bibcode=2014NatCo...5.4213P }} It employs the variational method (quantum mechanics), which guarantees that the expectation value of the Hamiltonian for any parameterized trial wave function is at least the lowest energy eigenvalue of that Hamiltonian.{{Cite journal |last1=Chan |first1=Albie |last2=Shi |first2=Zheng |last3=Dellantonio |first3=Luca |last4=Dur |first4=Wolfgang |last5=Muschik |first5=Christine A |date=2024 |title=Measurement-Based Infused Circuits for Variational Quantum Eigensolvers |journal=Physical Review Letters |volume=132 |issue=24 |page=240601 |doi=10.1103/PhysRevLett.132.240601 |arxiv=2305.19200 }} VQE is a hybrid algorithm that utilizes both quantum and classical computers. The quantum computer prepares and measures the quantum state, while the classical computer processes these measurements and updates the system. This synergy allows VQE to overcome some limitations of purely quantum methods.{{Cite journal |last1=Tilly |first1=Jules |last2=Chen |first2=Hongxiang |last3=Cao |first3=Shuxiang |last4=Picozzi |first4=Dario |last5=Setia |first5=Kanav |last6=Li |first6=Ying |last7=Grant |first7=Edward |last8=Wossnig |first8=Leonard |last9=Rungger |first9=Ivan |last10=Booth |first10=George H. |last11=Tennyson |first11=Jonathan |date=2022-11-05 |title=The Variational Quantum Eigensolver: A review of methods and best practices |url=https://www.sciencedirect.com/science/article/pii/S0370157322003118 |journal=Physics Reports |volume=986 |pages=1–128 |doi=10.1016/j.physrep.2022.08.003 |arxiv=2111.05176 |bibcode=2022PhR...986....1T |s2cid=243861087 |issn=0370-1573}}

{{see also|Density matrix}}

The reduced density matrices (1-RDM and 2-RDM) can be used to extrapolate the electronic structure of a system.{{Cite journal |last1=Liu |first1=Jie |last2=Li |first2=Zhenyu |last3=Yang |first3=Jinlong |date=2021-06-28 |title=An efficient adaptive variational quantum solver of the Schrödinger equation based on reduced density matrices |journal=The Journal of Chemical Physics |volume=154 |issue=24 |doi=10.1063/5.0054822 |pmid=34241330 |arxiv=2012.07047 |bibcode=2021JChPh.154x4112L |s2cid=229156865 |issn=0021-9606}}

In the Hamiltonian variational ansatz, the initial state $|\backslash psi\_0\backslash rangle$  is prepared to represent the ground state of the molecular Hamiltonian without electron correlations. The evolution of this state under the Hamiltonian, split into commuting segments $H\_j$ , is given by the equation below.

$|\backslash psi(\backslash theta)\backslash rangle\; =\; \backslash prod\_d\; \backslash prod\_j\; e^\{i\backslash theta\_\{d,j\}\; H\_j\}\; |\backslash psi\_0\backslash rangle$

where $\backslash theta\_\{d,j\}$  are variational parameters optimized to minimize the energy, providing insights into the electronic structure of the molecule.

McClean et al. (2016) and Romero et al. (2019) proposed a formula to estimate the number of measurements ( $N\_m$ ) required for energy precision. The formula is given by $N\_m\; =\; \backslash left(\backslash sum\_i\; |h\_i|\backslash right)^2/\backslash epsilon^2$ , where $h\_i$ are coefficients of each Pauli string in the Hamiltonian. This leads to a scaling of $\backslash mathcal\{O\}(M^6/\backslash epsilon^2)$ in a Gaussian orbital basis and $\backslash mathcal\{O\}(M^4/\backslash epsilon^2)$ in a plane wave dual basis. Note that $M$ is the number of basis functions in the chosen basis set.{{Cite journal |last1=Romero |first1=Jonathan |last2=Babbush |first2=Ryan |last3=McClean |first3=Jarrod R |last4=Hempel |first4=Cornelius |last5=Love |first5=Peter J |last6=Aspuru-Guzik |first6=Alán |date=2018-10-19 |title=Strategies for quantum computing molecular energies using the unitary coupled cluster ansatz |journal=Quantum Science and Technology |volume=4 |issue=1 |page=014008 |doi=10.1088/2058-9565/aad3e4 |arxiv=1701.02691 |s2cid=4175437 |issn=2058-9565}}{{Cite journal |last1=McClean |first1=Jarrod R |last2=Romero |first2=Jonathan |last3=Babbush |first3=Ryan |last4=Aspuru-Guzik |first4=Alán |date=2016-02-04 |title=The theory of variational hybrid quantum-classical algorithms |journal=New Journal of Physics |volume=18 |issue=2 |page=023023 |doi=10.1088/1367-2630/18/2/023023 |arxiv=1509.04279 |bibcode=2016NJPh...18b3023M |s2cid=92988541 |issn=1367-2630}}

A method by Bonet-Monroig, Babbush, and O'Brien (2019) focuses on grouping terms at a fermionic level rather than a qubit level, leading to a measurement requirement of only $\backslash mathcal\{O\}(M^2)$ circuits with an additional gate depth of $\backslash mathcal\{O\}(M)$.{{Cite journal |last1=Bonet-Monroig |first1=Xavier |last2=Babbush |first2=Ryan |last3=O'Brien |first3=Thomas E. |date=2020-09-22 |title=Nearly Optimal Measurement Scheduling for Partial Tomography of Quantum States |url=https://link.aps.org/doi/10.1103/PhysRevX.10.031064 |journal=Physical Review X |volume=10 |issue=3 |page=031064 |doi=10.1103/PhysRevX.10.031064|arxiv=1908.05628 |bibcode=2020PhRvX..10c1064B |s2cid=199668962 }}

While VQE's application in solving the electronic Schrödinger equation for small molecules has shown success, its scalability is hindered by two main challenges: the complexity of the quantum circuits required and the intricacies involved in the classical optimization process.{{Cite journal |last1=Grimsley |first1=Harper R. |last2=Barron |first2=George S. |last3=Barnes |first3=Edwin |last4=Economou |first4=Sophia E. |last5=Mayhall |first5=Nicholas J. |date=2023-03-01 |title=Adaptive, problem-tailored variational quantum eigensolver mitigates rough parameter landscapes and barren plateaus |url=https://www.nature.com/articles/s41534-023-00681-0 |journal=npj Quantum Information |language=en |volume=9 |issue=1 |page=19 |doi=10.1038/s41534-023-00681-0 |arxiv=2204.07179 |bibcode=2023npjQI...9...19G |s2cid=257236255 |issn=2056-6387}} These challenges are significantly influenced by the choice of the variational ansatz, which is used to construct the trial wave function. Modern quantum computers face limitations in running deep quantum circuits, especially when using the existing ansatzes for problems that exceed several qubits.

{{main|Jordan-Wigner transformation}}

Jordan-Wigner encoding is a method in quantum computing used for simulating fermionic systems like molecular orbitals and electron interactions in quantum chemistry.{{Cite journal |last1=Jiang |first1=Zhang |last2=Sung |first2=Kevin J. |last3=Kechedzhi |first3=Kostyantyn |last4=Smelyanskiy |first4=Vadim N. |last5=Boixo |first5=Sergio |date=2018-04-26 |title=Quantum Algorithms to Simulate Many-Body Physics of Correlated Fermions |journal=Physical Review Applied |language=en |volume=9 |issue=4 |page=044036 |doi=10.1103/PhysRevApplied.9.044036 |bibcode=2018PhRvP...9d4036J |s2cid=54064506 |issn=2331-7019|doi-access=free |arxiv=1711.05395 }}

In quantum chemistry, electrons are modeled as fermions with antisymmetric wave functions. The Jordan-Wigner encoding maps these fermionic orbitals to qubits, preserving their antisymmetric nature. Mathematically, this is achieved by associating each fermionic creation $(\; a^\backslash dagger\_i\; )$ and annihilation $(\; a\_i\; )$ operator with corresponding qubit operators through the Jordan-Wigner transformation:

$a^\backslash dagger\_i\; \backslash rightarrow\; \backslash frac\{1\}\{2\}\; \backslash left(\; \backslash prod\_\{k=1\}^\{i-1\}\; Z\_k\; \backslash right)\; (X\_i\; -\; iY\_i)$

Where $X\_i$ , $Y\_i$ , and $Z\_i$ are Pauli matrices acting on the $i^\{\backslash text\{th\}\}$ qubit.

Electron hopping between orbitals, central to chemical bonding and reactions, is represented by terms like $a^\{\backslash dagger\_i\}\; a\_j\; +\; a^\{\backslash dagger\_j\}\; a\_i$

. Under Jordan-Wigner encoding, these transform as follows:$$a^\backslash dagger\_i\; a\_j\; +\; a^\backslash dagger\_j\; a\_i\; \backslash rightarrow\; \backslash frac\{1\}\{2\}\; (X\_i\; X\_j\; +\; Y\_i\; Y\_j)\; Z\_\{i+1\}\; \backslash cdots\; Z\_\{j-1\}$$This transformation captures the quantum mechanical behavior of electron movement and interaction within molecules.{{Cite journal |last1=Li |first1=Qing-Song |last2=Liu |first2=Huan-Yu |last3=Wang |first3=Qingchun |last4=Wu |first4=Yu-Chun |last5=Guo |first5=Guo-Ping |date=2022 |title=A unified framework of transformations based on the Jordan–Wigner transformation |url=https://pubs.aip.org/jcp/article/157/13/134104/2841832/A-unified-framework-of-transformations-based-on |access-date=2023-11-13 |journal=The Journal of Chemical Physics |volume=157 |issue=13 |doi=10.1063/5.0107546|pmid=36209000 |arxiv=2108.01725 |bibcode=2022JChPh.157m4104L |s2cid=236912625 }}

The complexity of simulating a molecular system using Jordan-Wigner encoding is influenced by the structure of the molecule and the nature of electron interactions. For a molecular system with $K$ orbitals, the number of required qubits scales linearly with $K$ , but the complexity of gate operations depends on the specific interactions being modeled.{{Cite arXiv |last1=Harrison |first1=Brent |last2=Nelson |first2=Dylan |last3=Adamiak |first3=Daniel |last4=Whitfield |first4=James |date=November 2022 |title=Reducing the qubit requirement of Jordan-Wigner encodings of N-mode, K-fermion systems from N to ⌈log2(NK)⌉ |class=quant-ph |eprint=2211.04501 }}

The Jordan-Wigner transformation encodes fermionic operators into qubit operators, but it introduces non-local string operators that can make simulations inefficient. The FSWAP gate is used to mitigate this inefficiency by rearranging the ordering of fermions (or their qubit representations), thus simplifying the implementation of fermionic operations.{{Cite web |title=Custom Fermionic Codes for Quantum Simulation {{!}} Perimeter Institute |url=https://www2.perimeterinstitute.ca/videos/custom-fermionic-codes-quantum-simulation |access-date=2023-11-13 |website=www2.perimeterinstitute.ca}}

FSWAP networks rearrange qubits to efficiently simulate electron dynamics in molecules. These networks are essential for reducing the gate complexity in simulations, especially for non-neighboring electron interactions.{{Cite journal |last1=Kivlichan |first1=Ian D. |last2=McClean |first2=Jarrod |last3=Wiebe |first3=Nathan |last4=Gidney |first4=Craig |last5=Aspuru-Guzik |first5=Alán |last6=Chan |first6=Garnet Kin-Lic |last7=Babbush |first7=Ryan |date=2018-03-13 |title=Quantum Simulation of Electronic Structure with Linear Depth and Connectivity |url=https://link.aps.org/doi/10.1103/PhysRevLett.120.110501 |journal=Physical Review Letters |volume=120 |issue=11 |page=110501 |arxiv=1711.04789 |bibcode=2018PhRvL.120k0501K |doi=10.1103/PhysRevLett.120.110501 |pmid=29601758 |s2cid=4219888}}

When two fermionic modes (represented as qubits after the Jordan-Wigner transformation) are swapped, the FSWAP gate not only exchanges their states but also correctly updates the phase of the wavefunction to maintain fermionic antisymmetry. This is in contrast to the standard SWAP gate, which does not account for the phase change required in the antisymmetric wavefunctions of fermions.{{Cite arXiv |eprint=2111.04572 |class=quant-ph |first1=Akel |last1=Hashim |first2=Rich |last2=Rines |title=Optimized fermionic SWAP networks with equivalent circuit averaging for QAOA |date=2021 |last3=Omole |first3=Victory |last4=Naik |first4=Ravi K |last5=John MarkKreikebaum |first5=John Mark |last6=Santiago |first6=David I |last7=Chong |first7=Frederic T. |last8=Siddiqi |first8=Irfan |last9=Gokhale |first9=Pranav}}

The use of FSWAP gates can significantly reduce the complexity of quantum circuits for simulating fermionic systems. By intelligently rearranging the fermions, the number of gates required to simulate certain fermionic operations can be reduced, leading to more efficient simulations. This is particularly useful in simulations where fermions need to be moved across large distances within the system, as it can avoid the need for long chains of operations that would otherwise be required.{{Cite journal |last1=Rubin |first1=Nicholas C. |last2=Gunst |first2=Klaas |last3=White |first3=Alec |last4=Freitag |first4=Leon |last5=Throssell |first5=Kyle |last6=Chan |first6=Garnet Kin-Lic |last7=Babbush |first7=Ryan |last8=Shiozaki |first8=Toru |date=2021-10-27 |title=The Fermionic Quantum Emulator |url=https://quantum-journal.org/papers/q-2021-10-27-568/ |journal=Quantum |language=en-GB |volume=5 |page=568 |arxiv=2104.13944 |bibcode=2021Quant...5..568R |doi=10.22331/q-2021-10-27-568 |s2cid=233443911}}

{{Reflist}}

Category:Quantum chemistry

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• {{Cite book|title=Handbook of computational quantum chemistry|last=Cook|first=David Branston|publisher=Oxford University Press|year=1998|isbn=9780198501145|oclc=468919475}}
• {{cite book

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| title = Quantum Chemistry in the Age of Machine Learning

| year = 2023

| publisher = Elsevier

| location = Amsterdam, Netherlands

| editor-last1 = Dral

| editor-first1 = Pavlo O.

|isbn=9780323900492|oclc=1294286017|lccn=2024443666}}

{{refend}}

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